Prime numbers, prime factors and composite numbers
In a nutshell
Prime and composite numbers are special categories of numbers based on how many factors they have. There are $25$ prime numbers and $74$ composite numbers between $1$ and $100$.
Prime numbers and composite numbers
Prime numbers only have two factors, $1$ and itself. Composite numbers have more than two factors.
Example 1
Identify whether the following numbers are prime or composite: $2,10,13,25$.
Number  Factors  Prime or Composite 
$2$  $1,2$  $Prime$ 
$10$  $1,2,5,10$  $Composite$ 
$13$  $1,13$  $Prime$ 
$25$  $1,5,25$  $Composite$ 
Note: $1$ is neither a prime or composite number because it only has one factor, $1$.
Prime factors
Factors of a number which are prime are prime factors. They can be found using a factor tree to breakdown a number into its prime factors which can be multiplied together to give the original number. This can be done by following the procedure below:
PROCEDURE
$1$
 Write down the given number at the top.

$2$
 Find any two numbers which multiply to give the top number. 
$3$
 Write down these two factors beneath the original number at the end of a left and right "branch". 
$4$
 Repeat this process for all the numbers in the tree until each "branch" ends in a prime factor.

$5$
 Circle each prime factor, and write them all multiplied by each other.

Example 2
What is $32$ as a product of its prime factors?
Write $32$ on the top, and find two numbers that multiply to give $32$:
$32=2\times16$
Write $2$ and $16$ beneath $32$. $2$ is a prime number so circle it.
${\begin{aligned} &\space \space \,\,\,\,32\\\ &\,\,\swarrow\searrow& \\ &\textcircled{2} \quad16 \end{aligned}}$
As $16$ is not a prime number, find two numbers that multiply to give $16$ such as: $2 \times 8$. Add this to the factor tree and circle $2$ since it is prime.
$\quad{\begin{aligned} &\space \space \,\,\,\,32\\\ &\,\,\swarrow\searrow& \\ &\textcircled{2} \quad16 \\ &\quad\,\,\swarrow\searrow\\ &\space\space\space\,\,\textcircled2 \,\,\quad8 \end{aligned}}$
As $8$ is not a prime number, find two numbers that multiply to give $8$ such as: $2 \times 4$. Add this to the factor tree and circle $2$ since it is prime.
$\quad\quad{\begin{aligned} &\space \space \,\,\,\,32\\\ &\,\,\swarrow\searrow& \\ &\textcircled{2} \quad16 \\ &\quad\,\,\swarrow\searrow\\ &\space\space\space\,\,\textcircled2 \,\,\quad8 \\ &\quad\quad\,\,\,\swarrow\searrow \\ &\quad\space\space\,\,\textcircled2 \quad\space\,\,\, 4 \end{aligned}}$
As $4$ is not a prime number, find two numbers that multiply to give $4$ such as $2 \times 2$. Add this to the factor tree and circle both $2$s since they are prime.
$\quad\quad\quad\space{\begin{aligned} &\space \space \,\,\,\,32\\\ &\,\,\swarrow\searrow& \\ &\textcircled{2} \quad16 \\ &\quad\,\,\swarrow\searrow\\ &\space\space\space\,\,\textcircled2 \,\,\quad8 \\ &\quad\quad\,\,\,\swarrow\searrow \\ &\quad\space\space\,\,\textcircled2 \quad\space\,\,\, 4 \\ &\quad\quad\quad\space\,\,\swarrow\searrow \\ &\quad\quad\quad\textcircled2 \quad\space \,\,\textcircled2\end{aligned}}$
Only prime factors remain so the tree is complete.
Write $32$ as a product of its prime factors.
Therefore, $\underline{2 \times 2\times 2\times 2\times 2=32.}$
Note: $1$ is not a prime number so it does not appear in the factor tree.